The R\'enyi entropy of L\'evy distribution

The equivalence between non-extensive C. Tsallis entropy and the extensive entropy introduced by Alfr\'ed R\'enyi is discussed. The R\'enyi entropy is studied from the perspective of the geometry of the Lebesgue and generalised, exotic Lebesgue spaces. A duality principle is established. The R\'enyi entropy for the L\'evy distribution, in the domain when the nunerical methods fails, is approximated by asymptotic expansion for the large values of the R\'enyi parameter.Comment: 10 pages, 0 figure

Minimum Dissipation Principle in Nonlinear Transport

We extend Onsager's minimum dissipation principle to stationary states that are only subject to local equilibrium constraints, even when the transport coefficients depend on the thermodynamic forces. Crucial to this generalization is a decomposition of the thermodynamic forces into those that are held fixed by the boundary conditions, and the subspace which is orthogonal with respect to the metric defined by the transport coefficients. We are then able to apply Onsager and Machlup's proof to the second set of forces. As an example we consider two-dimensional nonlinear diffusion coupled to two reservoirs at different temperatures. Our extension differs from that of Bertini et al. in that we assume microscopic irreversibility and we allow a nonlinear dependence of the fluxes on the forces.Comment: 20 pages, 1 figur

Symmetry Group and Group Representations Associated to the Thermodynamic Covariance Principle (TCP)

We describe the Lie group and the group representations associated to the nonlinear Thermodynamic Coordinate Transformations (TCT). The TCT guarantee the validity of the Thermodynamic Covariance Principle (TCP) : {\it The nonlinear closure equations, i.e. the flux-force relations, everywhere and in particular outside the Onsager region, must be covariant under TCT}. In other terms, the fundamental laws of thermodynamics should be manifestly covariant under transformations between the admissible thermodynamic forces, i.e. under TCT. The TCP ensures the validity of the fundamental theorems for systems far from equilibrium. The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. We derive the conserved (thermodynamic) currents and, as an example of calculation, a simple system out of equilibrium where the validity of TCP is imposed at the level of the kinetic equations is also analyzed.Comment: 35 pages, 6 figure

Relaxation of Chemical Reactions to Stationary States in the Chemical Affinities Space

Using the mass balance equations for chemical reactions, we show how the system relaxes towards a steady state in and out of the Onsager region. In the chemical affinities space, after fast transients, the relaxation process is a straight line when operating in the Onsager region, while out of this regime, the evolution of the system is such that the projections of the evolution equations for the forces and the shortest path on the flows coincide. For spatially-extended systems, similar results are valid for the evolution of the thermodynamic mode (i.e., the mode with wave-number k = 0). These results allow us to obtain the expression for the affine connection of the space covered by the thermodynamic forces, close to the steady states. Through the affine connection, the nonlinear closure equations are derived.Comment: 23 pages

New Class of Generalized Extensive Entropies for Studying Dynamical Systems in Highly Anisotropic Phase Space

Starting from the geometrical interpretation of the R\'enyi entropy, we introduce further extensive generalizations and study their properties. In particular, we found the probability distribution function obtained by the MaxEnt principle with generalized entropies. We prove that for a large class of dynamical systems subject to random perturbations, including particle transport in random media, these entropies play the role of Liapunov functionals. Some physical examples, which can be treated by the generalized R\'enyi entropies are also illustrated.Comment: 13 pages, 0 figure